StudyPebble

SAT Function Notation: What f(x) Means

Function notation is short and easy to misread. On the SAT you will evaluate a function at a value, work backward from an output to an input, and combine two functions in one expression. Understanding that f of 3 means substitute 3 for the variable, not multiply by 3, clears most of the difficulty immediately.

The rule

f of a number means substitute that number for the variable everywhere in the rule, never multiply by it.

The name of a function is just a label. In f of x equals 4x minus 7, the letter f is the rule and x is a placeholder for whatever goes in. Writing f of 3 means replace every x with 3, giving 12 minus 7, which is 5. The parentheses are not multiplication. They mark the input.

The placeholder can be anything, including another expression. If f of x equals 4x minus 7, then f of 2a equals 4 times 2a minus 7, which is 8a minus 7. And f of x plus 1 equals 4 times the quantity x plus 1 minus 7, which is 4x minus 3. The safest habit is to write the rule with empty parentheses where the x was, then drop the new input in: f of blank equals 4 times blank minus 7.

Questions often run the substitution backward. If f of x equals 4x minus 7 and the question says f of c equals 13, you set 4c minus 7 equal to 13 and solve, getting c equals 5. Notice the difference between this and the forward direction. Forward gives you an output, backward gives you an input, and the answer choices will contain both numbers.

Combining functions is a matter of evaluating each piece separately and then doing the arithmetic. Suppose g of x equals x squared plus 1 and h of x equals 2x. Then g of 3 minus 2 times h of 1 equals 10 minus 2 times 2, which is 6. Compute each function value first and write it down before combining, because the subtraction sign in front of a whole term is where sign errors happen.

Composition looks harder than it is. The expression f of g of 2 means evaluate the inner function first, then feed that result into the outer one. With g of x equals x squared plus 1 and f of x equals 4x minus 7, the inner g of 2 is 5, and f of 5 is 13. Work from the inside out every time, and label the intermediate value so you do not feed the wrong number forward.

The trap

Reading f of 3 as f times 3 and multiplying the whole expression, which produces an answer that is usually one of the choices.

Try it

Try one
Real question

The cost to rent a bike is $5 plus $3 per hour. After how many hours does the cost equal $26?

A city's recycling center processed 8,400 tons in its 3rd year of operation and 14,700 tons in its 10th year. Tonnage has grown at a constant rate. How many tons does the model predict for the 16th year?

  • A) 14,400
  • B) 19,200
  • C) 20,100 ✓
  • D) 21,000

Why: Tonnage rose 6,300 over 7 years, which is 900 tons per year. From year 10 to year 16 is 6 years, so add 5,400: 14,700 + 5,400 = 20,100. Choice D counts 7 years instead of 6. Answering 14,400 means you multiplied the annual growth by 16 and ignored the model's starting value. Answering 19,200 means you counted 12 years of growth between the 3rd and 16th years instead of 13. Answering 21,000 means you counted 7 years of growth after the 10th year instead of 6.

Practice more of these free

Questions students ask

What is the difference between f of x equals 5 and f of 5?

f of 5 is a question about the output when the input is 5, so you substitute. f of x equals 5 is an equation saying the output is 5, so you set the rule equal to 5 and solve for the input. The two give different numbers and both usually appear in the answer choices.

How do I read a function from a table on the SAT?

The first column is the input and the second is the output, so the row with x equal to 2 and f of x equal to 11 tells you f of 2 is 11. To answer a backward question, scan the output column for the given value and read the input beside it.

Guides that cover this

More on linear functions